A Robust Regression Method Based on Pearson Type VI Distribution

Abstract

AbstractIn classical regression analysis, the distribution of the error is assumed to be Gaussian, and Least Squares (LS) estimation method is used for parameter estimation. In practice, even if the distribution of errors is assumed to be Gaussian, residuals are not generally Gaussian. If the data set contains outlier (s) or there are observations that are suspected to be outlier, normality assumption is violated, and parameter estimates will be biased. Many statisticians used robust method, such as the M-Estimation Method, which is a generalized version of the Maximum Likelihood (ML) Estimation method, for parameter estimation when such problems occurred. However, if the data set has skewness and excess kurtosis, traditional M-Estimators cannot achieve a good solution. In this study, using the relationship between Pearson Differential Equation (PDE) and Influence Function (IF), M-Estimation method is proposed for data sets that follow Pearson Type VI (PVI) distribution. The advantage of this method takes into account the skewness and kurtosis values of the data set and generates dynamic solutions. Objective, influence, weight functions and tail properties of the PVI distribution are obtained by using the Probability Density Function (pdf) of the PVI distribution. For the regression parameter estimates, Iteratively Re-Weighted Least Squares Estimation Method (IRWLS) is used. In many simulation studies with different scenarios and applications with real data, if the data have skewness and excess kurtosis, the proposed method has achieved better results than other M-Estimation methods in terms of Total Absolute Deviation (TAB) and Mean Square Error (MSE).KeywordsM-Estimation methodRobust regressionPearson type VI distributionInfluence functionIteratively re-weighted least squares method